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Tata lectures on theta. III. (English) Zbl 0744.14033
Progress in Mathematics. 97. Boston, MA etc.: Birkhäuser. vii, 202 p. (1991).
This volume is the third of a three volume series on the theory of theta functions. Like the first volume (1983; Zbl 0509.14049) and the second one (1984; Zbl 0549.14014), it is based on D. Mumford’s lectures held at the Tata Institute of Fundamental Research (Bombay) during November 1978- -March 1979. Whilst Volume I and II are devoted to the classical, geometric and analytic theory of complex abelian varieties and their associated theta functions, together with their most recent applications to moduli spaces of curves and Jacobian varieties, non-linear partial differential equations and special integrable Hamiltonian systems, this final volume focusses on the purely algebraic theory of theta functions. The aim of the authors is to compare and clarify the deep interrelations between the different ways of viewing (and using) theta functions in analysis, algebraic geometry, and representation theory. To be more precise, theta functions occur
(i) as analytic functions depending on complex vectors and Riemann matrices (in the classical setting),
(ii) as matrix coefficients of representations of Heisenberg groups and/or metaplectic groups, and
(iii) as sections of line bundles on abelian varieties and/or their moduli spaces (in the general setting of algebraic geometry over arbitrary groundfields),
and these three conceptually and methodically completely different frameworks for theta functions are shown to be equivalent, on an ultimate and deep level, in the course of the book.
In this regard, the present third volume, consisting of the concluding fourth chapter of the whole series, reflects the recent progress that has been made, to a great extent by the authors themselves, in understanding the ubiquity and crucial role of theta functions in various branches of mathematics. - In particular, the algebraic aspect of theta functions, which is the relevant one when viewing them as sections of line bundles on families of polarized abelian varieties over arbitrary base schemes, has been introduced and developed about 25 years ago by D. Mumford in his celebrated three part paper “On the equations defining abelian varieties” published in Invent. Math. 1, 287-354 (1966), 3, 75-135 and 215-244 (1967; Zbl 0219.14024). As this paper is highly advanced and barely accessible to non-specialists, and since the development of this general (algebraic) approach is still far from being completed, the authors also give a simplified explicit treatment of the algebraic theory of theta functions, including D. Mumford’s basic original ideas and, moreover, related further results by G. Kempf, I. Barsotti, J.-I. Igusa, L. Moret-Bailly and the co-autor P. Norman. The structure of the book is as follows:
Section 1 provides an introduction to Heisenberg groups. These are locally compact complex Lie groups which contain the unit circle $$\mathbb{C}^*_ 1$$ as a normal subgroup, in the center, such that the factor group is abelian and locally compact. The central result, within this introduction, is the main theorem about representations of Heisenberg groups, which is due to Stone, Von Neumann, and Mackey. — In section 2 the theory of Heisenberg representations is specialized to the real case, that is, to the case where the factor group of a Heisenberg group is a real vector space. This includes the construction of special realizations of such Heisenberg representations via elements in the Siegel upper-half space, particularly the so-called Fock representation, and the introduction of theta functions as matrix coefficients for these special realizations. — Section 3 deals with the interrelation between the theta functions associated with Heisenberg representations, and the classical ones defined by complex tori. This link is explained by showing how the representation theory of finite Heisenberg groups occurs, in a natural way, in the study of sections of line bundles on complex abelian varieties.
In section 4 the authors turn to the purely algebraic theory of theta functions, and introduce the reader to the adelic methods developed by D. Mumford in his original approach to that topic. In the course of this section, they discuss the $$p$$-adic (adelic) version of Heisenberg groups and their significance for the study of towers of abelian varieties, isogenies and (symmetric) line bundles on abelian varieties. This is then used, in section 5, to define theta functions algebraically, namely as certain functions on the tower of an abelian variety, which are associated with sections of ample symmetric line bundles on the corresponding abelian variety. The fact that the construction and the basic properties of these algebraic theta functions are quite parallel to the classical analytical theory, is very skillfully demonstrated. In an appendix I to section 5, a scheme-theoretical version (with a view towards relative abelian schemes) of the algebraic theta functions is briefly sketched, and the following appendix II provides a panorama relating all the important Heisenberg representations.
Section 6 is devoted to a further generalization of theta functions, namely to the construction of theta series associated with positive definite (rational) quadratic forms. This is used, in the sequel, to study the algebra of theta functions, in particular the polynomial identities satisfied by them. Among those theta relations which are of protruding importance, is the algebraic generalization of Riemann’s theta relation. Its special reformulations and interpretations, mainly in terms of the Heisenberg action on towers of sections of ample degree one symmetric line bundles on abelian varieties, are fully explained in section 7, whereas in the following section 8 the classical functional equation (with respect to points in the Siegel upper-half space) of Riemann’s theta function is generalized to the algebraic theta functions characterized as matrix coefficients of the representations of the real Heisenberg groups. Then, in section 9, a further very natural and important generalization of the theta functions is investigated. The authors introduce theta functions depending not only on quadratic forms, as in section 6, but also on a fixed spherical harmonic polynomial. This is done from three different points of view, namely
(i) by differentiating analytic theta functions with respect to polynomial partial differential operators,
(ii) by the representation-theoretic treatment of theta functions and their transformation laws with respect to pluri-harmonic functions and, finally,
(iii) by using the purely algebraic method of defining analytic modular forms (as theta functions) for pluri-harmonic polynomials and (positive definite rational) quadratic forms, which is essentially due to I. Barsotti [cf. Sympos. Math., Roma 3, 247-277 (1970; Zbl 0194.522)].
The very general theory of algebraic theta functions explained in this section leads directly to the frontiers of current research on moduli theory for general abelian schemes [cf., e.g. L. Moret-Bailly, “Pinceaux de variétés abéliennes”, Astérisque 129 (1985; Zbl 0595.14032)].
The concluding section 10 is devoted to one of the main applications of theta functions in algebraic geometry, namely to the study of the homogeneous coordinate ring of an abelian variety. The topic discussed here originated with D. Mumford’s earlier work on the explicit computation of bases for linear systems on abelian varieties [cf. Lectures at C.I.M.E. $$3^ 0$$ Ciclo Varenna, 1969, Quest. algebraic Varieties, 29-100 (1970; Zbl 0198.258) and the paper in Invent. Math. 1 and 3 cited above]. Subsequently, these questions have been investigated by S. Koizumi, T. Sekiguchi, G. R. Kempf, and others. Two approaches have proved to be far-reaching: the direct study of linear systems by theta functions and the more abstract cohomological methods together with the use of the finite Heisenberg group (cf. section 3 of this book). Following closely the recent work by G. R. Kempf [cf. Am. J. Math. 111, No. 1, 65-94 (1989; Zbl 0673.14023)], the authors demonstrate here how these two basic methods work. The book ends with some related comments on the projective embeddings of the moduli spaces for abelian varieties with level-$$n$$ structures.
Altogether, this third volume represents the point of culmination of this extremely beautiful and important series of D. Mumford’s lectures. It throws a bridge, unique in literature, between the classical theory of theta functions, as developed in the first two volumes, and the most recent ideas, generalizations and developments in the current research. In the last few sections many open problems are raised, and the entire fascinating, enlightening and masterly presentation of that highly advanced topic should be appealing to both experts in the field and ambitious (graduate) students, just as to physicists working in quantum field theory.
This book is an indispensible guide to the current literature!

##### MSC:
 14K25 Theta functions and abelian varieties 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14K10 Algebraic moduli of abelian varieties, classification 11F27 Theta series; Weil representation; theta correspondences